Woods, who is also deputy governor of the Bank of England, said the regulator has already identified two key risks including the extra costs and risks firms may incur through fragmentation caused by Brexit and the disruption to the servicing of existing contracts and obstacles to data transfer.
The chief executive of the Prudential Regulation Authority (PRA) has warned new Treasury committee leader Nicky Morgan that a cliff-edge Brexit could force the regulator into "difficult prioritisation decisions".
Woods cautioned that fragmentation of market-based finance could result in higher costs and less activity, and highlighted the UK's withdrawal from the European Union as having the potential to affect the economy.
Banks, and other financial services firms, are likely to set up new subsidiaries on either side of the Channel as an insurance policy against a hard Brexit, in which United Kingdom firms lose access to Europe's single market.
Woods added: "This could require banks to be able to withstand, and continue lending in, an environment of higher loan impairments, increased risk of default and lower asset prices and collateral values".
In his reply, Woods said the PRA was scrutinising the plans of 401 City firms to ensure each one was ready for Brexit and also to establish iwhether there were "broader financial stability risks, which could arise from the collective execution of the contingency plans".
In particular in relation to "outbound" firms, "restructurings will in many cases result in strongly inter-connected entities between the United Kingdom and the European Union", he noted.
He said the PRA "may have to make some hard prioritisation decisions in order to accommodate" the extra work stemming from the UK's exit.
Woods added that risks to financial stability will deepen due to Brexit, as he gave his backing to securing a transitional exit deal. However, firms have begun to reveal how they intend to cope with the UK's exit from the EU.
"I have also asked Mr Woods for his views on the desirability and design of a transitional arrangement with the EU, to provide more time to negotiate and prepare for a new UK-EU economic relationship".
According to Woods, this would make such firms more complex and hard to supervise because of strong UK-EU interconnections.
Morgan said: "Based on the information the PRA has collected, I have asked Sam Woods about how banks and insurers will respond as the Brexit deadline approaches, and the key risks of a no deal scenario".
In the banking sector, for instance, about 7o firms operate this way, so they may need to gain authorisation from the PRA if they want to keep doing business in the UK.
The regulator has received more than 400 responses to a call for contingency plans from the firms it supervises. It has warned it may need to ask for more if the biggest risks around Brexit crystallise.